Abstract

The design of an off-axis mirror system, based on a pair of prolate spheroids with a common focus but not a common axis, is considered. With the focus as a pupil, any ray through one focus of a spheroid must pass through all others as each such ray becomes a chief ray. A pseudo- (optical) axis is defined as that chief ray about which other rays are symmetric. A condition to ensure the existence of the pseudoaxis is derived based on a relation between the two eccentricities, the angle that the pseudoaxis makes with the axis of the first spheroid and the angle between the axes of the two spheroids. Data are presented to compare theoretical calculations with computerized real ray-tracing data. A brief discussion is presented to consider the problem of imaging formation.

© 1992 Optical Society of America

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References

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  1. W. T. Welford, Aberrations of Optical Systems (Adam-Hilger, Boston, 1968), p. 210.
  2. D. Korsch, Reflective Optics (Academic, Boston, 1991), p. 105.
    [Crossref]
  3. H. P. Brueggeman, Conic Mirrors (Focal, London, 1968), p. 19.
  4. J. M. Simon, M. C. Simon, “The limits of validity of the plate diagram in off-axis systems,” Opt. Acta 25, 153–156 (1978).
    [Crossref]
  5. G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers, 2nd ed. (McGraw-Hill, New York, 1968), p. 48.
  6. D. J. Struik, Lectures on Classical Differential Geometry, 2nd ed. (Addison-Wesley, Reading, Mass., 1961).
  7. O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972).

1978 (1)

J. M. Simon, M. C. Simon, “The limits of validity of the plate diagram in off-axis systems,” Opt. Acta 25, 153–156 (1978).
[Crossref]

Brueggeman, H. P.

H. P. Brueggeman, Conic Mirrors (Focal, London, 1968), p. 19.

Korn, G. A.

G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers, 2nd ed. (McGraw-Hill, New York, 1968), p. 48.

Korn, T. M.

G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers, 2nd ed. (McGraw-Hill, New York, 1968), p. 48.

Korsch, D.

D. Korsch, Reflective Optics (Academic, Boston, 1991), p. 105.
[Crossref]

Simon, J. M.

J. M. Simon, M. C. Simon, “The limits of validity of the plate diagram in off-axis systems,” Opt. Acta 25, 153–156 (1978).
[Crossref]

Simon, M. C.

J. M. Simon, M. C. Simon, “The limits of validity of the plate diagram in off-axis systems,” Opt. Acta 25, 153–156 (1978).
[Crossref]

Stavroudis, O. N.

O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972).

Struik, D. J.

D. J. Struik, Lectures on Classical Differential Geometry, 2nd ed. (Addison-Wesley, Reading, Mass., 1961).

Welford, W. T.

W. T. Welford, Aberrations of Optical Systems (Adam-Hilger, Boston, 1968), p. 210.

Opt. Acta (1)

J. M. Simon, M. C. Simon, “The limits of validity of the plate diagram in off-axis systems,” Opt. Acta 25, 153–156 (1978).
[Crossref]

Other (6)

G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers, 2nd ed. (McGraw-Hill, New York, 1968), p. 48.

D. J. Struik, Lectures on Classical Differential Geometry, 2nd ed. (Addison-Wesley, Reading, Mass., 1961).

O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972).

W. T. Welford, Aberrations of Optical Systems (Adam-Hilger, Boston, 1968), p. 210.

D. Korsch, Reflective Optics (Academic, Boston, 1991), p. 105.
[Crossref]

H. P. Brueggeman, Conic Mirrors (Focal, London, 1968), p. 19.

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Figures (3)

Fig. 1
Fig. 1

Ellipse defined in terms of polar coordinates. The origin is at the proximal focus, as defined by Eqs. (1)(5). The vertex radius of curvature r is also the semilatus rectum. A ray at angle ϕ is illustrated as passing between the foci. The inclination of the normal is δ, and i is the angle of incidence and reflection.

Fig. 2
Fig. 2

Two coupled spheroids. The distal focus of the first spheroid coincides with the proximal focus of the second. The angle between the two axes of symmetry is β. The angles ϕ, ϕ′, ϕ″, and ϕ‴ are the angles that a continuous ray makes with the axes.

Fig. 3
Fig. 3

Pseudoaxis. Shown are two coupled spheroids. The angle between the pseudoaxis and the axis of the first spheroid is α, while ϕ0‴ is the angle between the pseudoaxis and the axis of the second spheroid. A secondary ray, at angle θ with respect to the pseudoaxis at the proximal focus of the first spheroid, exits at an angle ψ‴ with respect to the pseudoaxis at the distal focus of the second spheroid.

Tables (3)

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Table 1 Identification of the Pseudoaxisa

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Table 2 Calculation of the Pseudoaxisa

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Table 3 Confirmation of the Pseudoaxisa

Equations (43)

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ρ = r 1 - cos ϕ .
ρ = r 1 - cos ϕ ,
z 1 = r 1 - ,
z 2 = r 1 + .
d = z 1 - z 2 = 2 r 1 - 2 .
y = r sin ϕ 1 - cos ϕ = r sin ϕ 1 - cos ϕ , z = r cos ϕ 1 - cos ϕ = d - r cos ϕ 1 - cos ϕ ,
tan ϕ = ( 1 - 2 ) sin ϕ 2 - ( 1 + 2 ) cos ϕ .
sin ϕ = ( 1 - 2 ) sin ϕ / K 2 , cos ϕ = [ 2 - ( 1 + 2 ) cos ϕ ] / K 2 ,
K 2 = ( 1 + 2 ) - 2 cos ϕ .
ρ = r K 2 ( 1 - 2 ) ( 1 - cos ϕ ) = ρ K 2 ( 1 - 2 ) .
ρ + ρ = 2 r 1 - 2 .
tan δ = sin ϕ cos ϕ - .
tan i = sin ϕ 1 - cos ϕ ,
sin i = sin ϕ / K , cos i = ( 1 - cos ϕ ) / K ,
tan ϕ = ( 1 - 1 2 ) sin ( α + θ ) 2 1 - ( 1 + 1 2 ) cos ( α + θ ) ,
sin ϕ = ( 1 - 1 2 ) sin ( α + θ ) / K 1 2 , cos ϕ = [ 2 1 - ( 1 + 1 2 ) cos ( α + θ ) ] / K 1 2 ,
K 1 2 = ( 1 + 1 2 ) - 2 1 cos ( α + θ ) .
tan ϕ = ( 1 - 2 2 ) sin ( ϕ - β ) 2 2 - ( 1 + 2 2 ) cos ( ϕ - β ) ,
tan ϕ = ( 1 - 2 2 ) - 2 A + B * sin ( α + θ ) + C * cos ( α + θ ) 2 D - E * sin ( α + θ ) + F * cos ( α + θ ) ,
A = 1 sin β , B * = ( 1 - 1 2 ) cos β , C * = ( 1 + 1 2 ) sin β , D = 2 ( 1 + 1 2 ) - 1 ( 1 + 2 2 ) cos β , E * = ( 1 - 1 2 ) ( 1 + 2 2 ) sin β , F * = ( 1 + 1 2 ) ( 1 + 2 2 ) cos β - 4 1 2 .
tan ϕ = ( 1 - 2 2 ) - 2 A + B sin θ + C cos θ 2 D - E sin θ + F cos θ ,
B = B * cos α - C * sin α , C = B * sin α + C * cos α , E = E * cos α + F * sin α , F = - E * sin α + F * cos α .
tan ϕ 0 = ( 1 - 2 2 ) - 2 A + C 2 D + F .
tan ψ = tan ϕ - tan ϕ 0 1 + tan ϕ tan ϕ 0 .
tan ψ = ( 1 - 2 2 ) - P ( 1 - cos θ ) + Q sin θ 2 S - T sin θ + U cos θ ,
P = 2 ( A F + C D ) , Q = B ( 2 D + F ) - E ( 2 A - C ) , S = D ( 2 D + F ) + A ( 2 A - C ) ( 1 - 2 2 ) 2 , T = E ( 2 D + F ) + B ( 2 A - C ) ( 1 - 2 2 ) 2 , U = F ( 2 D + F ) - C ( 2 A - C ) ( 1 - 2 2 ) 2 .
A F + C D = 0 , E ( 2 D + F ) + B ( 2 A - C ) ( 1 - 2 2 ) 2 = 0.
A = - D E B ( 1 - 2 2 ) 2 , F = 1 E B C ( 1 - 2 2 ) 2 .
tan α = 2 ( 1 - 1 2 ) sin β 1 ( 1 + 2 2 ) - 2 ( 1 + 1 2 ) cos β .
p 2 = ( 1 + 1 2 2 2 ) - 2 1 2 cos β , q 2 = ( 1 2 + 2 2 ) - 2 1 2 cos β .
sin α = 2 ( 1 - 1 2 ) sin β / p q , cos α = [ 1 ( 1 + 2 2 ) - 2 ( 1 + 1 2 ) cos β ] / p q .
B = - ( 1 - 1 2 ) D / p q , C = 1 ( p 2 + q 2 ) sin β / p q , E = 1 ( 1 - 1 2 ) ( 1 - 2 2 ) 2 sin β / p q , F = - ( p 2 + q 2 ) D / p q ,
Q = ( 1 - 1 2 ) ( p - q ) 2 H / p q , S = - ( p - q ) 2 H / p q , U = ( p 2 + q 2 ) ( p - q ) 2 H / p 2 q 2 ,
H = D 2 + 1 2 ( 1 - 2 2 ) 2 sin 2 β .
tan ψ = - p q ( 1 - 1 2 ) ( 1 - 2 2 ) sin θ ( p 2 + q 2 ) cos θ - 2 p q .
( 1 - 1 2 ) ( 1 - 2 2 ) = p 2 - q 2 ,
tan ψ = p q ( p 2 - q 2 ) sin θ ( p 2 + q 2 ) cos θ - 2 p q .
tan ϕ 0 = - 1 ( 1 - 2 2 ) sin β / D ,
tan ( ϕ - β ) = 1 ( p - q ) 2 sin β 2 ( 1 - 1 2 ) 2 - 1 ( p - q ) 2 cos β .
1 ρ σ * = - 1 K ρ = - 1 - cos ϕ K r 1 ρ ϕ * = - 1 ρ ( 1 - cos ϕ ) 2 K 3 = - 1 r ( 1 - cos ϕ k ) 3 .
1 k σ = - 1 k σ + 2 cos i ρ σ * , cos 2 i k ϕ = - cos 2 i k ϕ + 2 cos i ρ ϕ * ,
1 k σ = - 1 k σ + 2 r ( 1 - cos ϕ K ) 2 , 1 k ϕ = - 1 k ϕ + 2 r ( 1 - cos ϕ K ) 2 .
1 t = - 1 t + 2 r ( 1 - cos ϕ K ) 2 .

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